Numerical solutions of Volterra integral equations with a solution dependent delay (Q1069690)

The authors consider second kind Volterra integral equations with a solution dependent delay: \[ y(x)=f(x)+\int^{x}_{0}H(x,t,y(t),y(t- \tau (y(t))))dt. \] They begin with the study of existence and uniqueness of solutions. In order to solve this equation numerically one has to combine methods for Volterra integral equations with such for delay differential equations. The authors use a quadrature formula (or product integration) for the discretization of the integral, and they apply an interpolation formula (e.g. cubic splines) to obtain an approximation to the solution between grid points. The convergence of the resulting method is investigated and two numerical examples (with known exact solution) are presented.